Quantum Mechanics-II

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Chapter#3: Time Dependent Perturbation Theory

All exercise S.Qs of Chapter#3: Time Dependent Perturbation Theory of Quantum Mechanics-II. We have arranged all S.Qs for BS/MSc Physics students.

Q.3.1 why transition probability at \omega_fi=\omega act as absorption for harmonic perturbation?

Answer:

\omega_{fi} -\ \omega=0

Ef\ -Ei =\hbar\omega

The perturbation can induce transitions from Ei only to a level Ef whose energy is higher than EI by just . Such a transition may be described as absorption energy \hbar\omega by the system.

Q.3.2 why energy is conserved in \delta(Ef-\ Ei) in constant perturbation?

Answer: The delta term  guarantees the conservation of energy: in the limit t\rightarrow\infty, the transition rate is non-vanishing only between states of equal energy. Hence, a constant (time independent) perturbation neither removes energy from the system nor supplies energy to it. It simply causes energy-conserving transitions.

Q.3.3 what is the time period of transition probability of oscillating sinusoidal function?

Answer:

    \[P_{fi}\left(t\right)=\frac{v^2_{fi}}{\hbar} \left[\frac{{4sin}^{2\ }\ (\omega_{fi\ }-\omega)}{(\omega_{fi\ }+\omega)\ ^2}\ \frac{t}{2}\right]\]

Or

    \[P_{fi}(t)=\left | -\frac{i}{\hbar}\int_{0}^{t}< \psi _{f}|\hat{V}(t')|\psi _i> e^{i\omega fit'}dt' \right |^2\]

This is time period.

Q.3.4 why absorption and emission process cannot take place simultaneously?

Answer: The absorption and emission are inverse process.

Absorption process:

When atoms absorbs energy of photon to promote electron to higher energy orbital;

Emission process:

When atoms emits energy as photon as electron falls from higher energy orbital to lower;

Q.3.5 wrote down necessary conditions for the existence of Pif for a harmonic perturbation.

Answer: The transition rate (\frac{P_{if} }{t}=

The first condition E_f=E_{i}- \hbar \omega is called emission.

The second condition E_f=\ E_{i\ } + \hbar \omega is called absorption.

Q.3.6 why transition probability has an interference pattern?

Answer: As a function of \omega_{fi} , the transition probability has an interference pattern. It is appreciable only near  \omega_{fi} = 0 and decays moves away from zero.

Q.3.7 Define MASER and LASER.

Answer: MASER: (Microwave amplification by stimulated emission of radiation).

LASER: (Light amplification by stimulated emission of radiation).

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